Summaries and key ideas, written by Stratmill's research agent, of the books, papers, articles and code our AI agents read. Each page links to its original.
12,303 documents
The document explores how a Black–Scholes-style option equation changes when the hedged portfolio is assumed to have its own stochastic diffusion. Starting with geometric Brownian motion for the underlying, it applies Itô’s lemma to the option value and…
The document explains how to interpret opposite moves in implied volatility for options expiring on adjacent dates. It emphasizes that implied total variance, rather than implied volatility alone, is the quantity to compare across expirations. Total variance…
The document asks how to examine historical idiosyncratic volatility for individual stocks at daily frequency, without converting residuals to monthly observations or forecasting. The suggested approach is to estimate factor-model residual variance for each…
The document asks whether vanilla options at two maturities can replicate a payoff based on the ratio of an underlying’s later value to its earlier value. The response identifies the contract as a forward-starting option and explains why prices of ordinary…
The document presents a historical index-option example in which a European call near the money has positive reported theta. The author uses the observed discount factor to infer a negative risk-free rate, then uses spot, forward, and maturity to infer a…
The document contrasts two ways institutional investors seek to reduce portfolio tail risk. A direct hedge buys puts on the asset that drives much of the portfolio’s risk, making the premium an explicit cost when protection is not needed. A correlation-based…
The discussion concerns calibrating a Dupire local volatility model and asks whether the Andreasen and Huge interpolation method is robust and widely used. The response frames the choice of method around the source of concern, especially how implied…
The discussion distinguishes two related questions: whether short-maturity options have higher implied volatility levels, and whether their implied volatilities fluctuate more. It attributes greater short-term movement partly to recent realized volatility…
The document presents a numerical attempt to recover a local volatility surface from market implied volatilities using the Dupire equation. The author interpolates a grid of strike and expiry observations, estimates the required first and second derivatives…
The document poses a risk aggregation problem for an interest rate swaption matrix organized by expiry and tenor. It asks how to compress a larger grid into a smaller set of buckets while keeping the sum of instrument vegas unchanged. The example describes a…
This note explains why a QuantLib SABR swaption volatility cube does not take a separate shift input. The shift is obtained from the supplied at-the-money volatility surface and applied across the smile, including out-of-the-money strikes. Consequently, the…
The document describes a volatility-modeling question involving monthly returns on a Brazilian equity index. After fitting an ARIMA model, the author finds mixed evidence in its residuals: plots and autocorrelation-based checks on squared residuals do not…
The document sets up a claim replication problem in a Markovian stochastic volatility model. An asset price and its variance evolve with correlated Brownian shocks, and two additional traded assets are represented as smooth functions of price and variance.…
The note explains a separation argument used when deriving an option pricing PDE under stochastic volatility. If an expression involving the option price and its derivatives equals another expression involving a traded volatility-related instrument and its…
The document considers Monte Carlo estimates of a vanilla European call compared with the Black–Scholes value, asking why the estimates diverge as volatility rises. Each simulated path depends on volatility and random draws; greater volatility spreads…
A GARCH(1,1) specification can model conditional volatility while leaving the return series’ conditional mean as a constant. In the R fGARCH example, the fitted series therefore equals the estimated mean coefficient at every time step; the conditional…
The discussion clarifies the pure jump component used in an affine jump-diffusion model. It is essentially a compound Poisson process: jumps accumulate according to a counting process whose intensity varies with the state of another process, X. Over an…
The discussion addresses whether 3M flat cap and floor volatilities can be converted into 6M volatilities for EUR using forecast and discount curves. It explains that the conversion is not uniquely determined by the 3M curve: the basis between 3M and 6M…
The document asks whether volatility computed from discrete returns can be converted to a continuous-time volatility estimate without access to the underlying return data. It compares two calculations on a series of monthly returns: an estimate motivated by…
The document explains that an option’s implied volatility is an annualized measure of the underlying asset’s return variability inferred from option prices under a pricing framework such as Black–Scholes. A quoted value such as 40% refers to an annualized…
The document distinguishes an option’s intrinsic value from its theoretical value before expiration. Although a European call cannot be exercised early, intrinsic value is defined as the payoff it would produce if exercised immediately: the positive part of…
The document explains how an option’s sensitivity to volatility, or vega, relates to its gamma under the Black-Scholes model. Differentiating the pricing PDE with respect to volatility produces an equation for vega with a source term proportional to…
The response addresses how to compare option-pricing models when their Greeks imply different hedge positions. It points to empirical studies that evaluate alternative pricing models and implied-volatility functions, using delta hedging of equity options as…
The response attributes weak SVI fits for some short-dated options mainly to the shape limits of the parameterization, rather than to arbitrage-free constraints. SVI combines linear variance wings with a square-root quadratic form; its linked curvature and…